Schôichi Ôta

Derivations on algebras of unbounded operators

This paper is a study of derivations on unbounded operator algebras in connection with those in operator algebras. In particular we study spatiality of derivations in several situations. We give the characterization of derivations on general *-algebras by using positive linear functionals. We also show that a derivation with some range-property on a left EW#-algebra induced by an unbounded Hilbert algebra is strongly implemented by an operator which belongs to an algebra of measurable operators.

Closed linear operators with domain containing their range

In connection with algebras of unbounded operators, Lassner showed in [4] that, if T is a densely defined, closed linear operator in a Hilbert space such that its domain is contained in the domain of its adjoint T * and is globally invariant under T and T *,then T is bounded. In the case of a Banach space (in particular, a C *-algebra) weshowed in [6] that a densely defined closed derivation in a C *-algebra with domaincontaining its range is automatically bounded (see the references in [6] and [7] for thetheory of derivations in C *-algebras).